On Unsheared Tetrads (original) (raw)
Related papers
Summary of Tetragrammaton Papers
2020
The following summary includes my monograph titles. Please go to my Academia site and view the actual papers that are referenced in this summary. Carl Monroe Elam
Generalization of the tetrad representation theorem
1993
The tetrad representation theorem, due to Spirtes, Glymour, and Scheines (1993), gives a graphical condition necessary and su cient for the vanishing of tetrad di erences in a linear correlation structure. This note simpli es their proof and generalizes the ...
Pacific Journal of Mathematics, 1985
An Abelian group G is called cotorsion-free if 0 is the only pure-injective subgroup contained in G. If G is a cotorsion-free Abelian group, we construct a slender, fr^-free Abelian group A such that Hom(Λ, G) = 0. This will be used to answer some questions about radicals and torsion theories of Abelian groups. 0. Introduction. In this paper we will consider torsion free abelian groups from I. Kaplansky's point of view: "In this strange part of the subject anything that can conceivably happen actually does happen", cf. [K, p. 81]. This statement which is supported by classical results holds in an even more spectacular sense which was not expected at this time. There are many results on torsion free abelian groups which are undecidable in ZFC, the axioms of Zermelo-Frankel set theory including the axiom of choice. The first suφrising result of this kind after years of stagnation was Shelah's solution of the famous Whitehead problem [SI]. In this paper Shelah also constructed for the first time arbitrarily large indecomposable abelian groups, thus improving classical results of S. Pontrjagin, R. Baer, I. Kaplansky, L. Fuchs, A. L. S. Corner and others, compare [Fu2, Vol. II] and [K]. Indecomposable abelian groups are necessarily cotorsion-free with only a few exceptions. These are the cyclic groups of prime power Z p «, the Prϋfer groups Z(/?°°), the group of rational numbers Q and the additive group J p of /?-adic integers. A group is called cotorsion-free if and only if it contains only the trivial cotorsion subgroup 0, cf. [GW1]. Remember that C is cotorsion (in the sense of K. H. Harrison) if Ext z (Q, C) = 0. From simple properties of cotorsion groups we conclude that a group G is cotorsion-free if and only if G is torsion-free (Z p £G),reduced(Q£G)andJp£ G), reduced (Q £ G) and J p £G),reduced(Q£G)andJpt G for all primes p, cf. [GW1]. For countable groups cotorsion-free is the same as reduced and torsion-free. A. L. S. Corner's celebrated theorem indicates then that each ring with a countable and cotorsion-free additive structure is the endomorphism ring of some (cotorsion-free) abelian group, cf. [Ful, Vol. II]. This result was extended by the authors [DG2] to arbitrary rings with cotorsion-free additive groups which are then realized on arbitrarily large cotorsion-free abelian groups. Using rings without non-trivial idempotents, indecomposable groups of 79 80 MANFRED DUGAS AND RUDIGER GOBEL any size can be obtained and the aforementioned result becomes a trivial consequence of [DG2]. However, using other elementary ring constructions this result supports Kaplansky's point of view in many aspects, e.g. there are many new different counter examples for I. Kaplansky's test problems. Similar results which are in many cases even undecidable im ZFC have been derived in [DG1], [EM], [Me], [DH1] and others. One of the questions "close" to results undecidable in ZFC is related with "rigid systems". A class {A^ i e /} of abelian groups is semi-rigid if Hom(v4 /? Aj) Φ 0 Φ Hom(Aj, A t) implies i = j for any /, j e /. This class is rigid if already Hom(yl 2 , Aj) Φ 0 implies i = j. The class is proper if / is not a set. M. Dugas and S. Herden [DH1] constructed proper rigid classes of (indecomposable) abelian groups using GδdePs axiom of constructibility V = L. Such a result cannot be expected in ZFC alone as follows from the Vopenka principle. However, at least semi-rigid proper classes exist in ZFC as recently shown by R. Gόbel and S. Shelah [GS]. This result is based on a construction of arbitrarily large cotorsion-free abelian groups A with the property that U = A for any subgroup U c A with \U\ = \A\ and A/U cotorsion free. All these constructions are highly sophisticated using transfinite induction on generating elements. The very heart of this paper is a similar kind of result based on a much simpler construction. Due to the elementary construction of the groups (4.2) we are able to pose stronger conditions on their structure, which allow us to answer some open problems and give new solutions to some already settled problems. These extra conditions are the properties N 1-free and slender. A group is called S Γ free if all its countable subgroups are free. The most popular non-free S 1-free groups are products Z* of the integers, in particular the Baer-Specker group Z s°. The proof that Z*° is fc^-free and not free is due to R. Baer and E. Specker, cf. [Ful, Vol. I]. We will use R. J. Nunke's well-known characterization of slender groups as a definition. Hence a group is slender if and only if it is cotorsion free and if it does not contain a copy of the Baer-Specker group. Then we have the following quite powerful
On Non-Cayley Tetravalent Metacirculant Graphs
Graphs and Combinatorics, 2002
In connection with the classification problem for non-Cayley tetravalent metacirculant graphs, three families of special tetravalent metacirculant graphs, denoted by U 1 , U 2 and U 3 , have been defined . It has also been shown [11] that any non-Cayley tetravalent metacirculant graph is isomorphic to a union of disjoint copies of a non-Cayley graph in one of the families U 1 , U 2 or U 3 . A natural question raised from the result is whether all graphs in these families are non-Cayley. We have proved recently in [12] that every graph in U 2 is non-Cayley. In this paper, we show that every graph in U 1 is also a connected non-Cayley graph and find an infinite class of connected non-Cayley graphs in the family U 3 .
On quasigroups rich in associative triples
Discrete Mathematics, 1983
Let G be a group and G(*) a quasigroup on the ~.~ame underlying set. Let dist(G, G(*)) denote the number of pairs (x,y)eG 2 such that xy~x*y. For a finite quasig4"oup Q, n = card(Q), let t = dist(Q) = min dist(G, Q), where G ruu-~ through all groups with the same underlying set, and s = s(Q) the number of non-associative triples. Then 4tn-2t 2 -24t ~ s ~< 4m. If 1~,.~<3n2/32, then 3m<s holds as well. Let n~>168 be an even integer and let tr=mins(Q), where Q runs through all non-associative quasigroups of order n. Then tr= 16n -64.
On torsion torsionfree triples
2007
A mi familia y amigos. A Patri. Contents Introducción (Spanish) i 65 4.1.1. Motivation 65 4.1.2. Outline of the chapter 65 i ii CONTENTS INTRODUCCIÓN (SPANISH) iii 3) Existen isomorfismos de anillos (DA)(P, P ) ∼ = C y (DA)(Q, Q) ∼ = B. 4) (DA)(P [n], Q) = 0 para todo n ∈ Z. 5) Si M es un complejo de D − A tal que (DA)(P [n], M ) = 0 y (DA)(Q[n], M ) = 0 para todo n ∈ Z, entonces M = 0.
Tetrads of lines spanning PG(7,2)
Bulletin of the Belgian Mathematical Society Simon Stevin, 2012
Our starting point is a very simple one, namely that of a set L_4 of four mutually skew lines in PG(7,2): Under the natural action of the stabilizer group G(L_4) < GL(8,2) the 255 points of PG(7,2) fall into four orbits omega_1, omega_2, omega_3 omega_4; of respective lengths 12, 54, 108, 81: We show that the 135 points in omega_2 \cup omega_4 are the internal points of a hyperbolic quadric H_7 determined by L_4; and that the 81-set omega_4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to (Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a triplet of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with the recent finding that each Segre S = S_3(2) in PG(7,2) determines a distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S" of S.
A Family of Tetravalent Half-transitive Graphs
2020
In this paper, we introduce a new family of graphs, Gamma(n,a)\Gamma(n,a)Gamma(n,a). We show that it is an infinite family of tetravalent half-transitive Cayley graphs. Apart from that, we determine some structural properties of Gamma(n,a)\Gamma(n,a)Gamma(n,a).
Discrete Mathematics 35 (198 1) 25-38 North-Holland Publishing Company
An explicit construction is given Which produces all the proper fiats and the'Tutte ~lynomial of a geometric lattice (or, more generally, a matroid) when only the hypeq&nes are know. A further construction explicitly ccrlculatzs the polychromate (a genera&&on of the Tutte polynomial) for a graph from its vertex-deleted subgrqhs.